Nonlinear Equations for Finite-Amplitude Wave Propagation in Fiber-Reinforced Hyperelastic Media

Size: px
Start display at page:

Download "Nonlinear Equations for Finite-Amplitude Wave Propagation in Fiber-Reinforced Hyperelastic Media"

Transcription

1 Nonlinear Equations for Finite-Amplitude Wave Propagation in Fiber-Reinforced Hyperelastic Media Alexei F. Cheviakov Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada December 7, 2014 A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

2 Collaborators J.-F. Ganghoffer, LEMTA - ENSEM, Université de Lorraine, Nancy, France Simon St. Jean, M.Sc. graduate, University of Saskatchewan A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

3 Notation Notation u x ux. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

4 Examples Collagen fiber in connective biological tissue (cnx.org/content/m46049/latest/) A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

5 Examples Arterial tissue (Holzapfel, Gasser, and Ogden, 2000) A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

6 Examples Fabric A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

7 Examples Appropriate framework: incompressible hyperelasticity. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

8 Notation; Material Picture A.F. Cheviakov, J.-F. Ganghoffer / J. Math. Anal. Appl. 396 (2012) Fig. 1. Material and Eulerian coordinates. osition x of a material point labeled by X Ω 0 at time t is given by Material picture, t), x i = φ i (X, t). A solid body occupies the reference (Lagrangian) volume Ω 0 R 3. in the reference configuration are commonly referred to as Lagrangian coordinates, and ac Actual (Eulerian) configuration: Ω R oordinates. The deformed body occupies an 3. Eulerian domain Ω = φ(ω 0 ) R 3 (Fig. 1). T X is given Material by points are labeled by X Ω 0. = dx dt The dφ actual dt. position of a material point: x = x (X, t) Ω. g φ must be sufficiently smooth (the regularity conditions depending on the particular proble A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

9 Notation; Material Picture A.F. Cheviakov, J.-F. Ganghoffer / J. Math. Anal. Appl. 396 (2012) Fig. 1. Material and Eulerian coordinates. osition x of a material point labeled by X Ω 0 at time t is given by Material picture, t), x i = φ i (X, t). in the Velocity reference ofconfiguration a material point arex: commonly v (X, t) = dx referred dt. to as Lagrangian coordinates, and ac oordinates. Jacobian Thematrix deformed (deformation body occupies gradient): an Eulerian domain Ω = φ(ω 0 ) R 3 (Fig. 1). T X is given by = dx dt dφ dt. F(X, t) = φ; J = det F > 0; g φ must be sufficiently smooth (the regularity conditions depending on the particular proble A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

10 Notation; Material Picture A.F. Cheviakov, J.-F. Ganghoffer / J. Math. Anal. Appl. 396 (2012) Fig. 1. Material and Eulerian coordinates. osition x of a material point labeled by X Ω 0 at time t is given by Material picture, t), x i = φ i (X, t). Boundary force (per unit area) in Eulerian configuration: t = σn. in the reference configuration are commonly referred to as Lagrangian coordinates, and ac Boundary force (per unit area) in Lagrangian configuration: T = PN. oordinates. The deformed body occupies an Eulerian domain Ω = φ(ω 0 ) R 3 (Fig. 1). T X is given σ = by σ(x, t) is the Cauchy stress tensor. = dx dt P = dφ JσF T dt. is the first Piola-Kirchhoff tensor. g φ must be sufficiently smooth (the regularity conditions depending on the particular proble A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

11 Notation; Material Picture A.F. Cheviakov, J.-F. Ganghoffer / J. Math. Anal. Appl. 396 (2012) Fig. 1. Material and Eulerian coordinates. osition x of a material point labeled by X Ω 0 at time t is given by Material picture, t), x i = φ i (X, t). Density in reference configuration: ρ 0 = ρ 0(X) (time-independent). in the reference configuration are commonly referred to as Lagrangian coordinates, and ac Density in actual configuration: oordinates. The deformed body occupies an Eulerian domain Ω = φ(ω 0 ) R 3 (Fig. 1). T X is given by ρ = ρ(x, t) = ρ 0/J. = dx dt dφ dt. g φ must be sufficiently smooth (the regularity conditions depending on the particular proble A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

12 Governing Equations Variables: Independent: time t, Lagrangian coordinates X Ω 0. Dependent: x = x(x, t), p = p(x, t), ρ = ρ(x, t). A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

13 Governing Equations Variables: Independent: time t, Lagrangian coordinates X Ω 0. Dependent: x = x(x, t), p = p(x, t), ρ = ρ(x, t). Incompressibility: J = det F = x i X j = 1, ρ = ρ0/j = ρ0(x). A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

14 Governing Equations Variables: Independent: time t, Lagrangian coordinates X Ω 0. Dependent: x = x(x, t), p = p(x, t), ρ = ρ(x, t). Incompressibility: J = det F = x i X j = 1, ρ = ρ0/j = ρ0(x). Equations of motion: ρ 0x tt = div (X ) P + ρ 0R, J = 1. R = R(X, t): total body force per unit mass. (div (X ) P) i = Pij X j. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

15 Constitutive Relations The first Piola-Kirchhoff stress tensor (incompressible): P ij = p (F 1 ) ji + ρ 0 W F ij, (1) W = W (X, F): a scalar strain energy density; p = p (X, t): hydrostatic pressure. Strain Energy Density W = W iso + W aniso. Isotropic Strain Energy Density For the left Cauchy-Green strain tensor B = FF T, I 1 = Tr B = F i kf i k, I 2 = 1 [(Tr 2 B)2 Tr(B 2 )], I 3 = det B = J 2 = 1. (2) Mooney-Rivlin materials: W iso = a(i 1 3) + b(i 2 3), a, b > 0, A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

16 Constitutive Relations Anisotropic Strain Energy Density In the reference configuration, fibers are oriented along the unit vector A = A(X). Actual fiber direction: a = a(x, t) = FA/ FA = FA/λ; λ = FA is the fiber stretch factor. Fiber invariants: I 4 = A T CA, I 5 = A T C 2 A. General model: W aniso = f (I 4 1, I 5 1), f (0, 0) = 0. Various specific models are used. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

17 Full Set of Equations Equations of motion: ρ 0x tt = div (X ) P, [ ] x i J = det = 1, X j P ij = p (F 1 ) ji + ρ 0 W F ij. 4 PDEs, 4 unknowns. Specific constitutive relation: W = W iso + W aniso = a(i 1 3) + b(i 2 3) + q (I 4 1) 2 ; a, b, q > 0. Isotropic part: Mooney-Rivlin-type; Anisotropic part: quadratic reinforcing model. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

18 Ansatz 1 Compatible with Incompressibility Equilibrium and Displacements Equilibrium/no displacement: x = X, natural state. Time-dependent, with displacement: x = X + G, G = G(X, t). No linearization/assumption of smallness of G. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

19 Ansatz 1 Compatible with Incompressibility Equilibrium and Displacements Equilibrium/no displacement: x = X, natural state. Time-dependent, with displacement: x = X + G, G = G(X, t). No linearization/assumption of smallness of G. Motions Transverse to a Plane x = X 1 X 2 X 3 + G ( X 1, X 2, t ), A = cos γ 0 sin γ. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

20 Ansatz 1 Compatible with Incompressibility Equilibrium and Displacements Equilibrium/no displacement: x = X, natural state. Time-dependent, with displacement: x = X + G, No linearization/assumption of smallness of G. Motions Transverse to a Plane x = X 1 X 2 X 3 + G ( X 1, X 2, t ), A = G = G(X, t). cos γ 0 sin γ. Deformation gradient: F = G/ X 1 G/ X 2 1, J = F 1. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

21 F = , G/ X 1 G/ X 2 1 Ansatz 1 Compatible with Incompressibility one has J 1, and the incompressibility condition is identically satisfied. X 3 A X 2 X 1 (a) 1 1 X 3 0 x X X 1-1 x x 1-1 (b) (c) (a) Fiber direction. Figure 1: (b) (a) Fiber Thedirection. reference (b)(lagrangian) The reference (Lagrangian) mesh with mesh fibers. with fibers. (c) A(c) sample A sample deformed mesh (Eulerian deformed configuration) mesh (Eulerian for configuration) this ansatz. for the ansatz (4.1). A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

22 Ansatz 1 Compatible with Incompressibility Current model Unknowns: G ( X 1, X 2, t ) and p ( X 1, X 2, t ). Parameters: Isotropic part: a, b = const > 0. Anisotropic part: q = const > 0. Fiber angle: γ. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

23 Ansatz 1 Compatible with Incompressibility Equations of motion: ( ) 2 G 2 G = 2 (a + b) 2 t (X 1 ) + 2 G ( 2 (X 2 ) 2 ( ) 2 G +4q cos 2 (γ) 3 cos 2 (γ) + 6 cos(γ) sin(γ) G X 1 X sin2 γ 0 = p ( G X + 1 2bρ0 X 1 2 G 8qρ 0 cos 3 γ (X 1 ) 2 0 = p ( G X + 2 2bρ0 X 2 2 G (X 2 ) G ) 2 G ( 2 X 2 X 1 X 2 cos γ G ) X + sin γ, 1 2 G (X 1 ) G 2 X 1 ) 2 G. X 1 X 2 ) 2 G (X 1 ) 2, A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

24 Ansatz 1 Compatible with Incompressibility Nonlinear wave equation with a differential constraint: Wave equation: ( ) 2 G 2 G = 2 (a + b) 2 t (X 1 ) + 2 G ( 2 (X 2 ) 2 ( ) 2 G +4q cos 2 (γ) 3 cos 2 (γ) + 6 cos(γ) sin(γ) G X 1 X sin2 γ Differential constraint: b [ G 2 G X 1 X 1 X 1 X G 2 X 2 = [ 4q cos 3 γ 2 G X 2 (X 1 ) 2 ] 2 G (X 1 ) 2 ( cos γ G X 1 + sin γ ) ( G + b X 2 ) 2 G X 1 X G 2 X 1 2 G (X 1 ) 2, )] 2 G (X 2 ) 2 A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

25 Ansatz 1 Compatible with Incompressibility Special case 1: fibers in X 3 -direction b X 1 [ G ( 2 G X 2 (X 1 ) G = 2 (a + b) 2 t ( 2 G (X 1 ) 2 + )] 2 G + b (X 2 ) 2 X 2 [ G X 1 2 G (X 2 ) 2 ), ( 2 G (X 1 ) 2 + )] 2 G = 0, (X 2 ) 2 No fiber effect. Here G = G(X 1, X 2, t). General fact: if the displacement does not depend on the fiber direction, i.e., ( G)A = 0, then I 4 J = const. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

26 Ansatz 1 Compatible with Incompressibility Special case 2: one-dimensional displacements X = X 1 X 2 X 3 + G ( X 1, t ) ; then (denote X 1 = x, G = G(x, t)) G tt = ( ( )) α + β cos 2 γ 3 cos 2 γ (G x) sin γ cos γg x + 2 sin 2 γ G xx, (no constraint). Pressure is given by p = βρ 0 cos 3 γ (cos γg x + 2 sin γ) G x + f (t). Here α = 2(a + b) > 0, β = 4q > 0. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

27 One Dimensional Transverse Waves Reference Configuration Actual Configuration A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

28 Symmetries of the Nonlinear Wave Equation PDE on G(x, t): G tt = ( α + β cos 2 γ ( )) 3 cos 2 γ (G x) sin γ cos γg x + 2 sin 2 γ G xx, Parameters arbitrary Symmetries Z 1 = x, Z2 = t, Z3 = G, Z4 = t G, Z 5 = x x + t t + G G 4α β, Z 1, Z 2, Z 3, Z 4, Z 5, ( ) cos 2 γ = 1 1 ± 1 4α Z 6 = 2t cos γ 2 β t + x cos γ x x sin γ G A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

29 Symmetries of the Nonlinear Wave Equation PDE on G(x, t): G tt = ( α + β cos 2 γ ( )) 3 cos 2 γ (G x) sin γ cos γg x + 2 sin 2 γ G xx, Parameters arbitrary Symmetries Z 1 = x, Z2 = t, Z3 = G, Z4 = t G, Z 5 = x x + t t + G G 4α β, Z 1, Z 2, Z 3, Z 4, Z 5, ( ) cos 2 γ = 1 1 ± 1 4α Z 6 = 2t cos γ 2 β t + x cos γ x x sin γ G The extra symmetry Z 6 can arise only for materials where the fiber contribution is sufficiently strong: β > 4α q > 2(a + b) (W = a(i 1 3) + b(i 2 3) + q (I 4 1) 2 ). A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

30 Remarks on the Nonlinear Wave Equation PDE on G(x, t): G tt = ( α + β cos 2 γ ( )) 3 cos 2 γ (G x) sin γ cos γg x + 2 sin 2 γ G xx, Remarks: Wave equations of the form u tt = F (u x)u xx can be linearized by a Hodograph transformation. Can be mapped by a point transformation into a constant-coefficient equation for a restricted class of F (u x). A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

31 Remarks on the Nonlinear Wave Equation PDE on G(x, t): G tt = ( α + β cos 2 γ ( )) 3 cos 2 γ (G x) sin γ cos γg x + 2 sin 2 γ G xx, Remarks: Wave equations of the form u tt = F (u x)u xx can be linearized by a Hodograph transformation. Can be mapped by a point transformation into a constant-coefficient equation for a restricted class of F (u x). PDE when γ = 0: transverse 1D waves propagating along the fibers ( G tt = α + 3β (G x) 2) G xx. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

32 1D Displacements Orthogonal to the Fibers Numerical solution G x p x A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

33 Ansatz 2 Compatible with Incompressibility Displacements transverse to an axis: X = X 1 X 2 + H ( X 1, t ) X 3 + G ( X 1, t ), A = cos γ 0 sin γ. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

34 Ansatz 2 Compatible with Incompressibility Displacements transverse to an axis: X = X 1 X 2 + H ( X 1, t ) X 3 + G ( X 1, t ), A = cos γ 0 sin γ. Deformation gradient: F = H/ X G/ X 1 0 1, J = F 1. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

35 Ansatz 2 Compatible with Incompressibility Displacements transverse to an axis: X = X 1 X 2 + H ( X 1, t ) X 3 + G ( X 1, t ), A = cos γ 0 sin γ. Deformation gradient: F = H/ X G/ X 1 0 1, J = F 1. Governing PDEs: Denote X 1 = x, G = G(x, t), H = H(x, t). A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

36 Ansatz 2 Compatible with Incompressibility Displacements transverse to an axis: X = X 1 X 2 + H ( X 1, t ) X 3 + G ( X 1, t ), A = cos γ 0 sin γ. Coupled nonlinear wave equations: 0 = p x 2βρ 0 cos 3 γ [(cos γg x + sin γ) G xx + cos γh xh xx], [ H tt = αh xx + β cos 3 γ cos γ ([ ] ] ) Gx 2 + Hx 2 Hxx + 2G xh xg xx + 2 sin γ x (GxHx), G tt = αg xx + β cos 2 γ [ 2 sin 2 γ G xx + cos 2 γ ( 2G xh xh xx + ( ) ) Hx 2 + 3Gx 2 Gxx + sin 2γ (3G xg xx + H xh xx)]. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

37 Ansatz 2 Compatible with Incompressibility Case 1: γ = π/2 H tt = αh xx, G tt = αg xx. Case 2: γ = 0 H tt = αh xx + β [([ 3H 2 x + G 2 x ] Hxx + 2G xh xg xx )], G tt = αg xx + β [( 2G xh xh xx + ( H 2 x + 3G 2 x ) Gxx )]. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

38 Ansatz 2 Compatible with Incompressibility Case 1: γ = π/2 H tt = αh xx, G tt = αg xx. Case 2: γ = 0 H tt = αh xx + β [([ 3H 2 x + G 2 x ] Hxx + 2G xh xg xx )], G tt = αg xx + β [( 2G xh xh xx + ( H 2 x + 3G 2 x ) Gxx )]. Traveling-wave solutions when γ = 0 r = x ct, H(x, t) = h(r), G(x, t) = g(r); [ α c 2 + β(3(h ) 2 + (g ) 2 ) ] h + 2βg h g = 0, 2βg h h + [ α c 2 + β((h ) 2 + 3(g ) 2 ) ] g = 0. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

39 Motions Transverse to an Axis: Traveling Wave Solutions Traveling-wave solutions when γ = 0 r = x ct, H(x, t) = h(r), G(x, t) = g(r); [ α c 2 + β(3(h ) 2 + (g ) 2 ) ] h + 2βg h g = 0, 2βg h h + [ α c 2 + β((h ) 2 + 3(g ) 2 ) ] g = 0. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

40 Motions Transverse to an Axis: Traveling Wave Solutions Traveling-wave solutions when γ = 0 r = x ct, H(x, t) = h(r), G(x, t) = g(r); [ α c 2 + β(3(h ) 2 + (g ) 2 ) ] h + 2βg h g = 0, 2βg h h + [ α c 2 + β((h ) 2 + 3(g ) 2 ) ] g = 0. Solutions: (g ) 2 + (h ) 2 = R 2 = c2 α, when c > c 0 = α = 2(a + b). β A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

41 Motions Transverse to an Axis: Traveling Wave Solutions Traveling-wave solutions when γ = 0 r = x ct, H(x, t) = h(r), G(x, t) = g(r); [ α c 2 + β(3(h ) 2 + (g ) 2 ) ] h + 2βg h g = 0, 2βg h h + [ α c 2 + β((h ) 2 + 3(g ) 2 ) ] g = 0. Solutions: (g ) 2 + (h ) 2 = R 2 = c2 α, when c > c 0 = α = 2(a + b). β Sample solution: a traveling periodic wave h(r) = A cos(kr + φ 0), g(r) = A sin(kr + φ 0), A = R/k. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

42 Sample solution: a traveling periodic wave An exact solution: h(r) = A cos(kr + φ 0), g(r) = A sin(kr + φ 0), A = R/k. This describes a time-periodic perturbation of the stress-free steady state, given by X 1 0 x = x (X, t) = X 2 + A cos(k[x 1 ct] + φ 0). X 3 Every material point follows a circle. sin(k[x 1 ct] + φ 0) Material lines along the X 1 direction, given by X 2 = const and X 3 = const, become helices parameterized by X 1 (figure below). A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

43 Sample solution: a traveling periodic wave (a) (b) AFigure time-periodic 5: Sometraveling material perturbation. lines for XMaterial 2 = const, lines Xalong 3 = const the Xin 1 direction, the reference givenconfiguration by (a XThe 2, Xsame 3 = const, lines are in the shown. actual configuration, parameterized by (5.22) (b). 6 Conclusions A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

44 Conclusions and Open Problems Current work A class of models of anisotropic hyperelastic fiber-reinforced materials is considered. Nonlinear wave equations are derived for ansätze compatible with incompressibility. Symmetry properties and exact solutions are being studied. Future/ongoing work Work with two fiber families. Consider setups more closely related to applications: A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

45 Some references Ciarlet, P. G. Mathematical Elasticity. Volume I: Three-dimensional Elasticity. Elsevier, Marsden, J. E. and Hughes, T. J. R. Mathematical Foundations of Elasticity. Dover, C.A. Basciano, C. Kleinstreuer. Invariants-based anisotropic constitutive models of healthy aneurysmal abdominal aortic wall. J. Biomech. Eng., 131:021009, G.A. Holzapfel, T.C. Gasser, R.W. Ogden. A new constitutive framework for arterial wall mechanics and a comparative study of material models. J. Elasticity, 61:1 48, Cheviakov, A.F., and Ganghoffer, J.-F. Symmetry Properties of Two-Dimensional Ciarlet-Mooney-Rivlin Constitutive Models in Nonlinear Elastodynamics. J. Math. An. App., A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

46 Some references Ciarlet, P. G. Mathematical Elasticity. Volume I: Three-dimensional Elasticity. Elsevier, Marsden, J. E. and Hughes, T. J. R. Mathematical Foundations of Elasticity. Dover, C.A. Basciano, C. Kleinstreuer. Invariants-based anisotropic constitutive models of healthy aneurysmal abdominal aortic wall. J. Biomech. Eng., 131:021009, G.A. Holzapfel, T.C. Gasser, R.W. Ogden. A new constitutive framework for arterial wall mechanics and a comparative study of material models. J. Elasticity, 61:1 48, Cheviakov, A.F., and Ganghoffer, J.-F. Symmetry Properties of Two-Dimensional Ciarlet-Mooney-Rivlin Constitutive Models in Nonlinear Elastodynamics. J. Math. An. App., Thank you for your attention! A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31

Natural States and Symmetry Properties of. Two-Dimensional Ciarlet-Mooney-Rivlin. Nonlinear Constitutive Models

Natural States and Symmetry Properties of. Two-Dimensional Ciarlet-Mooney-Rivlin. Nonlinear Constitutive Models Natural States and Symmetry Properties of Two-Dimensional Ciarlet-Mooney-Rivlin Nonlinear Constitutive Models Alexei Cheviakov, Department of Mathematics and Statistics, Univ. Saskatchewan, Canada Jean-François

More information

Symmetry Properties of Two-Dimensional Ciarlet-Mooney-Rivlin Constitutive Models in Nonlinear Elastodynamics

Symmetry Properties of Two-Dimensional Ciarlet-Mooney-Rivlin Constitutive Models in Nonlinear Elastodynamics Symmetry Properties of Two-Dimensional Ciarlet-Mooney-Rivlin Constitutive Models in Nonlinear Elastodynamics A. F. Cheviakov a, J.-F. Ganghoffer b a Department of Mathematics and Statistics, University

More information

A comparison of conservation law construction approaches for the two-dimensional incompressible Mooney-Rivlin hyperelasticity model

A comparison of conservation law construction approaches for the two-dimensional incompressible Mooney-Rivlin hyperelasticity model JOURNAL OF MATHEMATICAL PHYSICS 56, 505 (05) A comparison of conservation law construction approaches for the two-dimensional incompressible Mooney-Rivlin hyperelasticity model A. F. Cheviakov and S. St.

More information

Conservation Laws for Nonlinear Equations: Theory, Computation, and Examples

Conservation Laws for Nonlinear Equations: Theory, Computation, and Examples Conservation Laws for Nonlinear Equations: Theory, Computation, and Examples Alexei Cheviakov Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada Seminar February 2015

More information

Connecting Euler and Lagrange systems as nonlocally related systems of dynamical nonlinear elasticity

Connecting Euler and Lagrange systems as nonlocally related systems of dynamical nonlinear elasticity Arch. Mech., 63, 4, pp. 1 20, Warszawa 2011 Connecting Euler and Lagrange systems as nonlocally related systems of dynamical nonlinear elasticity G. BLUMAN 1), J. F. GANGHOFFER 2) 1) Mathematics Department

More information

User-Materials in ANSYS

User-Materials in ANSYS User-Materials in ANSYS Holzapfel-Model l lfor Soft Tissues Prof. Dr.-Ing A. Fritsch Possibilities of user programming ANSYS User Programmable Features (UPF) are capabilities you can use to write your

More information

A Recursion Formula for the Construction of Local Conservation Laws of Differential Equations

A Recursion Formula for the Construction of Local Conservation Laws of Differential Equations A Recursion Formula for the Construction of Local Conservation Laws of Differential Equations Alexei Cheviakov Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada December

More information

FETI Methods for the Simulation of Biological Tissues

FETI Methods for the Simulation of Biological Tissues SpezialForschungsBereich F 32 Karl Franzens Universita t Graz Technische Universita t Graz Medizinische Universita t Graz FETI Methods for the Simulation of Biological Tissues Ch. Augustin O. Steinbach

More information

Lectures on. Constitutive Modelling of Arteries. Ray Ogden

Lectures on. Constitutive Modelling of Arteries. Ray Ogden Lectures on Constitutive Modelling of Arteries Ray Ogden University of Aberdeen Xi an Jiaotong University April 2011 Overview of the Ingredients of Continuum Mechanics needed in Soft Tissue Biomechanics

More information

Full-field measurements and identification for biological soft tissues: application to arteries in vitro

Full-field measurements and identification for biological soft tissues: application to arteries in vitro Centre for Health Engineering CNRS UMR 5146 INSERM IFR 143 Prof. Stéphane Avril Full-field measurements and identification for biological soft tissues: application to arteries in vitro using single-gage

More information

Applications of Symmetries and Conservation Laws to the Study of Nonlinear Elasticity Equations

Applications of Symmetries and Conservation Laws to the Study of Nonlinear Elasticity Equations Applications of Symmetries and Conservation Laws to the Study of Nonlinear Elasticity Equations A Thesis Submitted to the College of Graduate Studies and Research in Partial Fulfillment of the Requirements

More information

Nonlinear Elasticity, Anisotropy, Material Stability and Residual stresses in Soft Tissue

Nonlinear Elasticity, Anisotropy, Material Stability and Residual stresses in Soft Tissue Nonlinear Elasticity, Anisotropy, Material Stability and Residual stresses in Soft Tissue R.W. Ogden Department of Mathematics, University of Glasgow Glasgow G 8QW, UK Email: rwo@maths.gla.ac.uk Home Page:

More information

COMPUTATION OF EIGENSTRESSES IN THREE-DIMENSIONAL PATIENT-SPECIFIC ARTERIAL WALLS

COMPUTATION OF EIGENSTRESSES IN THREE-DIMENSIONAL PATIENT-SPECIFIC ARTERIAL WALLS IS Computation - Biomechanics of eigenstresses and Computational in three-dimensional Modeling of Living patient-specific Tissue arterial walls XIII International Conference on Computational Plasticity.

More information

Alexei F. Cheviakov. University of Saskatchewan, Saskatoon, Canada. INPL seminar June 09, 2011

Alexei F. Cheviakov. University of Saskatchewan, Saskatoon, Canada. INPL seminar June 09, 2011 Direct Method of Construction of Conservation Laws for Nonlinear Differential Equations, its Relation with Noether s Theorem, Applications, and Symbolic Software Alexei F. Cheviakov University of Saskatchewan,

More information

You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.

You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. MATHEMATICAL TRIPOS Part III Thursday 1 June 2006 1.30 to 4.30 PAPER 76 NONLINEAR CONTINUUM MECHANICS Attempt FOUR questions. There are SIX questions in total. The questions carry equal weight. STATIONERY

More information

Constitutive models. Constitutive model: determines P in terms of deformation

Constitutive models. Constitutive model: determines P in terms of deformation Constitutive models Constitutive model: determines P in terms of deformation Elastic material: P depends only on current F Hyperelastic material: work is independent of path strain energy density function

More information

AN ANISOTROPIC PSEUDO-ELASTIC MODEL FOR THE MULLINS EFFECT IN ARTERIAL TISSUE

AN ANISOTROPIC PSEUDO-ELASTIC MODEL FOR THE MULLINS EFFECT IN ARTERIAL TISSUE XI International Conference on Computational Plasticity. Fundamentals and Applications COMPLAS XI E. Oñate, D.R.J. Owen, D. Peric and B. Suárez (Eds) AN ANISOTROPIC PSEUDO-ELASTIC MODEL FOR THE MULLINS

More information

International Journal of Pure and Applied Mathematics Volume 58 No ,

International Journal of Pure and Applied Mathematics Volume 58 No , International Journal of Pure and Applied Mathematics Volume 58 No. 2 2010, 195-208 A NOTE ON THE LINEARIZED FINITE THEORY OF ELASTICITY Maria Luisa Tonon Department of Mathematics University of Turin

More information

Conservation Laws of Surfactant Transport Equations

Conservation Laws of Surfactant Transport Equations Conservation Laws of Surfactant Transport Equations Alexei Cheviakov Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada Winter 2011 CMS Meeting Dec. 10, 2011 A. Cheviakov

More information

Iranian Journal of Mathematical Sciences and Informatics Vol.2, No.2 (2007), pp 1-16

Iranian Journal of Mathematical Sciences and Informatics Vol.2, No.2 (2007), pp 1-16 Iranian Journal of Mathematical Sciences and Informatics Vol.2, No.2 (2007), pp 1-16 THE EFFECT OF PURE SHEAR ON THE REFLECTION OF PLANE WAVES AT THE BOUNDARY OF AN ELASTIC HALF-SPACE W. HUSSAIN DEPARTMENT

More information

Conservation Laws: Systematic Construction, Noether s Theorem, Applications, and Symbolic Computations.

Conservation Laws: Systematic Construction, Noether s Theorem, Applications, and Symbolic Computations. Conservation Laws: Systematic Construction, Noether s Theorem, Applications, and Symbolic Computations. Alexey Shevyakov Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon,

More information

LOCKING AND STABILITY OF 3D TEXTILE COMPOSITE REINFORCEMENTS DURING FORMING

LOCKING AND STABILITY OF 3D TEXTILE COMPOSITE REINFORCEMENTS DURING FORMING 10th International Conference on Composite Science and Technology ICCST/10 A.L. Araújo, J.R. Correia, C.M. Mota Soares, et al. (Editors) IDMEC 015 LOCKING AND STABILITY OF 3D TEXTILE COMPOSITE REINFORCEMENTS

More information

Professor George C. Johnson. ME185 - Introduction to Continuum Mechanics. Midterm Exam II. ) (1) x

Professor George C. Johnson. ME185 - Introduction to Continuum Mechanics. Midterm Exam II. ) (1) x Spring, 997 ME85 - Introduction to Continuum Mechanics Midterm Exam II roblem. (+ points) (a) Let ρ be the mass density, v be the velocity vector, be the Cauchy stress tensor, and b be the body force per

More information

Nonlinear elasticity and gels

Nonlinear elasticity and gels Nonlinear elasticity and gels M. Carme Calderer School of Mathematics University of Minnesota New Mexico Analysis Seminar New Mexico State University April 4-6, 2008 1 / 23 Outline Balance laws for gels

More information

Fundamentals of Linear Elasticity

Fundamentals of Linear Elasticity Fundamentals of Linear Elasticity Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research of the Polish Academy

More information

Elasticity Model for Blood Vessel

Elasticity Model for Blood Vessel Elasticity Model for Blood Vessel Violaine Louvet To cite this version: Violaine Louvet. Elasticity Model for Blood Vessel. study of different models of the mechanical non-linear stress-strain behaviour

More information

1 Useful Definitions or Concepts

1 Useful Definitions or Concepts 1 Useful Definitions or Concepts 1.1 Elastic constitutive laws One general type of elastic material model is the one called Cauchy elastic material, which depend on only the current local deformation of

More information

3D constitutive modeling of the passive heart wall

3D constitutive modeling of the passive heart wall 3D constitutive modeling of the passive heart wall Thomas S.E. Eriksson Institute of Biomechanics - Graz University of Technology Department of Biophysics - Medical University of Graz SFB status seminar,

More information

3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1

3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is

More information

Constitutive Relations

Constitutive Relations Constitutive Relations Andri Andriyana, Ph.D. Centre de Mise en Forme des Matériaux, CEMEF UMR CNRS 7635 École des Mines de Paris, 06904 Sophia Antipolis, France Spring, 2008 Outline Outline 1 Review of

More information

SIMULATION OF MECHANICAL TESTS OF COMPOSITE MATERIAL USING ANISOTROPIC HYPERELASTIC CONSTITUTIVE MODELS

SIMULATION OF MECHANICAL TESTS OF COMPOSITE MATERIAL USING ANISOTROPIC HYPERELASTIC CONSTITUTIVE MODELS Engineering MECHANICS, Vol. 18, 2011, No. 1, p. 23 32 23 SIMULATION OF MECHANICAL TESTS OF COMPOSITE MATERIAL USING ANISOTROPIC HYPERELASTIC CONSTITUTIVE MODELS Tomáš Lasota*, JiříBurša* This paper deals

More information

This introductory chapter presents some basic concepts of continuum mechanics, symbols and notations for future reference.

This introductory chapter presents some basic concepts of continuum mechanics, symbols and notations for future reference. Chapter 1 Introduction to Elasticity This introductory chapter presents some basic concepts of continuum mechanics, symbols and notations for future reference. 1.1 Kinematics of finite deformations We

More information

The Non-Linear Field Theories of Mechanics

The Non-Linear Field Theories of Mechanics С. Truesdell-W.Noll The Non-Linear Field Theories of Mechanics Second Edition with 28 Figures Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest Contents. The Non-Linear

More information

Constitutive Modeling of Biological Soft Tissues

Constitutive Modeling of Biological Soft Tissues Constitutive Modeling of Biological Soft Tissues Attila P. Nagy 1, David J. Benson 1, Vikas Kaul 2, Mark Palmer 2 1 Livermore Software Technology Corporation, Livermore, CA 94551, USA 2 Medtronic plc,

More information

Spline-Based Hyperelasticity for Transversely Isotropic Incompressible Materials

Spline-Based Hyperelasticity for Transversely Isotropic Incompressible Materials Paper 260 Civil-Comp Press, 2012 Proceedings of the Eleventh International Conference on Computational Structures Technology, B.H.V. Topping, (Editor), Civil-Comp Press, Stirlingshire, Scotland Spline-Based

More information

Law of behavior very-rubber band: almost incompressible material

Law of behavior very-rubber band: almost incompressible material Titre : Loi de comportement hyperélastique : matériau pres[...] Date : 25/09/2013 Page : 1/9 Law of behavior very-rubber band: almost incompressible material Summary: One describes here the formulation

More information

Elements of Rock Mechanics

Elements of Rock Mechanics Elements of Rock Mechanics Stress and strain Creep Constitutive equation Hooke's law Empirical relations Effects of porosity and fluids Anelasticity and viscoelasticity Reading: Shearer, 3 Stress Consider

More information

Constitutive Relations

Constitutive Relations Constitutive Relations Dr. Andri Andriyana Centre de Mise en Forme des Matériaux, CEMEF UMR CNRS 7635 École des Mines de Paris, 06904 Sophia Antipolis, France Spring, 2008 Outline Outline 1 Review of field

More information

Continuum Mechanics and the Finite Element Method

Continuum Mechanics and the Finite Element Method Continuum Mechanics and the Finite Element Method 1 Assignment 2 Due on March 2 nd @ midnight 2 Suppose you want to simulate this The familiar mass-spring system l 0 l y i X y i x Spring length before/after

More information

Effects of Aging on the Mechanical Behavior of Human Arteries in Large Deformations

Effects of Aging on the Mechanical Behavior of Human Arteries in Large Deformations International Academic Institute for Science and Technology International Academic Journal of Science and Engineering Vol. 3, No. 5, 2016, pp. 57-67. ISSN 2454-3896 International Academic Journal of Science

More information

in this web service Cambridge University Press

in this web service Cambridge University Press CONTINUUM MECHANICS This is a modern textbook for courses in continuum mechanics. It provides both the theoretical framework and the numerical methods required to model the behavior of continuous materials.

More information

Classification of Prostate Cancer Grades and T-Stages based on Tissue Elasticity Using Medical Image Analysis. Supplementary Document

Classification of Prostate Cancer Grades and T-Stages based on Tissue Elasticity Using Medical Image Analysis. Supplementary Document Classification of Prostate Cancer Grades and T-Stages based on Tissue Elasticity Using Medical Image Analysis Supplementary Document Shan Yang, Vladimir Jojic, Jun Lian, Ronald Chen, Hongtu Zhu, Ming C.

More information

Conservation of mass. Continuum Mechanics. Conservation of Momentum. Cauchy s Fundamental Postulate. # f body

Conservation of mass. Continuum Mechanics. Conservation of Momentum. Cauchy s Fundamental Postulate. # f body Continuum Mechanics We ll stick with the Lagrangian viewpoint for now Let s look at a deformable object World space: points x in the object as we see it Object space (or rest pose): points p in some reference

More information

2.1 Strain energy functions for incompressible materials

2.1 Strain energy functions for incompressible materials Chapter 2 Strain energy functions The aims of constitutive theories are to develop mathematical models for representing the real behavior of matter, to determine the material response and in general, to

More information

Traction on the Retina Induced by Saccadic Eye Movements in the Presence of Posterior Vitreous Detachment

Traction on the Retina Induced by Saccadic Eye Movements in the Presence of Posterior Vitreous Detachment Traction on the Retina Induced by Saccadic Eye Movements in the Presence of Posterior Vitreous Detachment Colangeli E., Repetto R., Tatone A. and Testa A. Grenoble, 24 th October 2007 Table of contents

More information

Inverse Design (and a lightweight introduction to the Finite Element Method) Stelian Coros

Inverse Design (and a lightweight introduction to the Finite Element Method) Stelian Coros Inverse Design (and a lightweight introduction to the Finite Element Method) Stelian Coros Computational Design Forward design: direct manipulation of design parameters Level of abstraction Exploration

More information

Simple shear is not so simple

Simple shear is not so simple Simple shear is not so simple Michel Destrade ab, Jerry G. Murphy c, Giuseppe Saccomandi d, a School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway, University

More information

Introduction to Seismology Spring 2008

Introduction to Seismology Spring 2008 MIT OpenCourseWare http://ocw.mit.edu 1.510 Introduction to Seismology Spring 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 1.510 Introduction to

More information

COMPUTATIONAL MODELING OF SHAPE MEMORY MATERIALS

COMPUTATIONAL MODELING OF SHAPE MEMORY MATERIALS COMPUTATIONAL MODELING OF SHAPE MEMORY MATERIALS Jan Valdman Institute of Information Theory and Automation, Czech Academy of Sciences (Prague) based on joint works with Martin Kružík and Miroslav Frost

More information

Traction on the Retina Induced by Saccadic Eye Movements in the Presence of Posterior Vitreous Detachment

Traction on the Retina Induced by Saccadic Eye Movements in the Presence of Posterior Vitreous Detachment Traction on the Retina nduced by Saccadic Eye Movements in the Presence of Posterior Vitreous etachment Colangeli E.,1, Repetto R. 1,2, Tatone A. 1 and Testa A. 1 1 SAT - epartment of Engineering of Structures,

More information

CS 468. Differential Geometry for Computer Science. Lecture 17 Surface Deformation

CS 468. Differential Geometry for Computer Science. Lecture 17 Surface Deformation CS 468 Differential Geometry for Computer Science Lecture 17 Surface Deformation Outline Fundamental theorem of surface geometry. Some terminology: embeddings, isometries, deformations. Curvature flows

More information

Measurement of deformation. Measurement of elastic force. Constitutive law. Finite element method

Measurement of deformation. Measurement of elastic force. Constitutive law. Finite element method Deformable Bodies Deformation x p(x) Given a rest shape x and its deformed configuration p(x), how large is the internal restoring force f(p)? To answer this question, we need a way to measure deformation

More information

Continuum Mechanics and Theory of Materials

Continuum Mechanics and Theory of Materials Peter Haupt Continuum Mechanics and Theory of Materials Translated from German by Joan A. Kurth Second Edition With 91 Figures, Springer Contents Introduction 1 1 Kinematics 7 1. 1 Material Bodies / 7

More information

Biomechanics. Soft Tissue Biomechanics

Biomechanics. Soft Tissue Biomechanics Biomechanics cross-bridges 3-D myocardium ventricles circulation Image Research Machines plc R* off k n k b Ca 2+ 0 R off Ca 2+ * k on R* on g f Ca 2+ R0 on Ca 2+ g Ca 2+ A* 1 A0 1 Ca 2+ Myofilament kinetic

More information

Engineering Sciences 241 Advanced Elasticity, Spring Distributed Thursday 8 February.

Engineering Sciences 241 Advanced Elasticity, Spring Distributed Thursday 8 February. Engineering Sciences 241 Advanced Elasticity, Spring 2001 J. R. Rice Homework Problems / Class Notes Mechanics of finite deformation (list of references at end) Distributed Thursday 8 February. Problems

More information

(2) 2. (3) 2 Using equation (3), the material time derivative of the Green-Lagrange strain tensor can be obtained as: 1 = + + +

(2) 2. (3) 2 Using equation (3), the material time derivative of the Green-Lagrange strain tensor can be obtained as: 1 = + + + LAGRANGIAN FORMULAION OF CONINUA Review of Continuum Kinematics he reader is referred to Belytscho et al. () for a concise review of the continuum mechanics concepts used here. he notation followed here

More information

By drawing Mohr s circle, the stress transformation in 2-D can be done graphically. + σ x σ y. cos 2θ + τ xy sin 2θ, (1) sin 2θ + τ xy cos 2θ.

By drawing Mohr s circle, the stress transformation in 2-D can be done graphically. + σ x σ y. cos 2θ + τ xy sin 2θ, (1) sin 2θ + τ xy cos 2θ. Mohr s Circle By drawing Mohr s circle, the stress transformation in -D can be done graphically. σ = σ x + σ y τ = σ x σ y + σ x σ y cos θ + τ xy sin θ, 1 sin θ + τ xy cos θ. Note that the angle of rotation,

More information

Mathematical Tripos Part IA Lent Term Example Sheet 1. Calculate its tangent vector dr/du at each point and hence find its total length.

Mathematical Tripos Part IA Lent Term Example Sheet 1. Calculate its tangent vector dr/du at each point and hence find its total length. Mathematical Tripos Part IA Lent Term 205 ector Calculus Prof B C Allanach Example Sheet Sketch the curve in the plane given parametrically by r(u) = ( x(u), y(u) ) = ( a cos 3 u, a sin 3 u ) with 0 u

More information

UNIVERSITY OF CALGARY. Nonlinear Elasticity, Fluid Flow and Remodelling in Biological Tissues. Aleksandar Tomic A THESIS

UNIVERSITY OF CALGARY. Nonlinear Elasticity, Fluid Flow and Remodelling in Biological Tissues. Aleksandar Tomic A THESIS UNIVERSITY OF CALGARY Nonlinear Elasticity, Fluid Flow and Remodelling in Biological Tissues by Aleksandar Tomic A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

More information

Introduction to Continuum Mechanics

Introduction to Continuum Mechanics Introduction to Continuum Mechanics I-Shih Liu Instituto de Matemática Universidade Federal do Rio de Janeiro 2018 Contents 1 Notations and tensor algebra 1 1.1 Vector space, inner product........................

More information

3D Elasticity Theory

3D Elasticity Theory 3D lasticity Theory Many structural analysis problems are analysed using the theory of elasticity in which Hooke s law is used to enforce proportionality between stress and strain at any deformation level.

More information

A note on finite elastic deformations of fibre-reinforced non-linearly elastic tubes

A note on finite elastic deformations of fibre-reinforced non-linearly elastic tubes Arch. Mech., 67, 1, pp. 95 109, Warszawa 2015 Brief Note A note on finite elastic deformations of fibre-reinforced non-linearly elastic tubes M. EL HAMDAOUI, J. MERODIO Department of Continuum Mechanics

More information

A CONTINUUM MECHANICS PRIMER

A CONTINUUM MECHANICS PRIMER A CONTINUUM MECHANICS PRIMER On Constitutive Theories of Materials I-SHIH LIU Rio de Janeiro Preface In this note, we concern only fundamental concepts of continuum mechanics for the formulation of basic

More information

A Numerical Study of Finite Element Calculations for Incompressible Materials under Applied Boundary Displacements

A Numerical Study of Finite Element Calculations for Incompressible Materials under Applied Boundary Displacements A Numerical Study of Finite Element Calculations for Incompressible Materials under Applied Boundary Displacements A Thesis Submitted to the College of Graduate Studies and Research in Partial Fulfillment

More information

Chapter 2. Rubber Elasticity:

Chapter 2. Rubber Elasticity: Chapter. Rubber Elasticity: The mechanical behavior of a rubber band, at first glance, might appear to be Hookean in that strain is close to 100% recoverable. However, the stress strain curve for a rubber

More information

Nonlocally related PDE systems for one-dimensional nonlinear elastodynamics

Nonlocally related PDE systems for one-dimensional nonlinear elastodynamics J Eng Math (2008) 62:203 22 DOI 0.007/s0665-008-922-7 Nonlocally related PDE systems for one-dimensional nonlinear elastodynamics G. Bluman A. F. Cheviakov J.-F. Ganghoffer Received: 5 July 2006 / Accepted:

More information

Course No: (1 st version: for graduate students) Course Name: Continuum Mechanics Offered by: Chyanbin Hwu

Course No: (1 st version: for graduate students) Course Name: Continuum Mechanics Offered by: Chyanbin Hwu Course No: (1 st version: for graduate students) Course Name: Continuum Mechanics Offered by: Chyanbin Hwu 2011. 11. 25 Contents: 1. Introduction 1.1 Basic Concepts of Continuum Mechanics 1.2 The Need

More information

Continuum Mechanics. Continuum Mechanics and Constitutive Equations

Continuum Mechanics. Continuum Mechanics and Constitutive Equations Continuum Mechanics Continuum Mechanics and Constitutive Equations Continuum mechanics pertains to the description of mechanical behavior of materials under the assumption that the material is a uniform

More information

KINEMATICS OF CONTINUA

KINEMATICS OF CONTINUA KINEMATICS OF CONTINUA Introduction Deformation of a continuum Configurations of a continuum Deformation mapping Descriptions of motion Material time derivative Velocity and acceleration Transformation

More information

NDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16.

NDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16. CAVITY INSPECTION NDT&E Methods: UT VJ Technologies NDT&E Methods: UT 6. NDT&E: Introduction to Methods 6.1. Ultrasonic Testing: Basics of Elasto-Dynamics 6.2. Principles of Measurement 6.3. The Pulse-Echo

More information

Asymptotic Behavior of Waves in a Nonuniform Medium

Asymptotic Behavior of Waves in a Nonuniform Medium Available at http://pvamuedu/aam Appl Appl Math ISSN: 1932-9466 Vol 12, Issue 1 June 217, pp 217 229 Applications Applied Mathematics: An International Journal AAM Asymptotic Behavior of Waves in a Nonuniform

More information

Nonlinear Theory of Elasticity. Dr.-Ing. Martin Ruess

Nonlinear Theory of Elasticity. Dr.-Ing. Martin Ruess Nonlinear Theory of Elasticity Dr.-Ing. Martin Ruess geometry description Cartesian global coordinate system with base vectors of the Euclidian space orthonormal basis origin O point P domain of a deformable

More information

EXPERIMENTAL IDENTIFICATION OF HYPERELASTIC MATERIAL PARAMETERS FOR CALCULATIONS BY THE FINITE ELEMENT METHOD

EXPERIMENTAL IDENTIFICATION OF HYPERELASTIC MATERIAL PARAMETERS FOR CALCULATIONS BY THE FINITE ELEMENT METHOD Journal of KONES Powertrain and Transport, Vol. 7, No. EXPERIMENTAL IDENTIFICATION OF HYPERELASTIC MATERIAL PARAMETERS FOR CALCULATIONS BY THE FINITE ELEMENT METHOD Robert Czabanowski Wroclaw University

More information

MITOCW MITRES2_002S10nonlinear_lec15_300k-mp4

MITOCW MITRES2_002S10nonlinear_lec15_300k-mp4 MITOCW MITRES2_002S10nonlinear_lec15_300k-mp4 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources

More information

PEAT SEISMOLOGY Lecture 2: Continuum mechanics

PEAT SEISMOLOGY Lecture 2: Continuum mechanics PEAT8002 - SEISMOLOGY Lecture 2: Continuum mechanics Nick Rawlinson Research School of Earth Sciences Australian National University Strain Strain is the formal description of the change in shape of a

More information

Math background. Physics. Simulation. Related phenomena. Frontiers in graphics. Rigid fluids

Math background. Physics. Simulation. Related phenomena. Frontiers in graphics. Rigid fluids Fluid dynamics Math background Physics Simulation Related phenomena Frontiers in graphics Rigid fluids Fields Domain Ω R2 Scalar field f :Ω R Vector field f : Ω R2 Types of derivatives Derivatives measure

More information

In recent years anisotropic materials have been finding their way into aerospace applications traditionally

In recent years anisotropic materials have been finding their way into aerospace applications traditionally 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Confere 1-4 May 2006, Newport, Rhode Island AIAA 2006-2250 Anisotropic Materials which Can be Modeled by Polyconvex Strain Energy

More information

A monolithic FEM solver for fluid structure

A monolithic FEM solver for fluid structure A monolithic FEM solver for fluid structure interaction p. 1/1 A monolithic FEM solver for fluid structure interaction Stefan Turek, Jaroslav Hron jaroslav.hron@mathematik.uni-dortmund.de Department of

More information

07 - balance principles. me338 - syllabus balance principles balance principles. cauchy s postulate. cauchy s lemma -t

07 - balance principles. me338 - syllabus balance principles balance principles. cauchy s postulate. cauchy s lemma -t me338 - syllabus holzapfel nonlinear solid mechanics [2000], chapter 4, pages 131-161 1 2 cauchy s postulate cauchy s lemma cauchy s postulate stress vector t to a plane with normal n at position x only

More information

NONLINEAR CONTINUUM FORMULATIONS CONTENTS

NONLINEAR CONTINUUM FORMULATIONS CONTENTS NONLINEAR CONTINUUM FORMULATIONS CONTENTS Introduction to nonlinear continuum mechanics Descriptions of motion Measures of stresses and strains Updated and Total Lagrangian formulations Continuum shell

More information

Modelling Anisotropic, Hyperelastic Materials in ABAQUS

Modelling Anisotropic, Hyperelastic Materials in ABAQUS Modelling Anisotropic, Hyperelastic Materials in ABAQUS Salvatore Federico and Walter Herzog Human Performance Laboratory, Faculty of Kinesiology, The University of Calgary 2500 University Drive NW, Calgary,

More information

Continuum mechanics V. Constitutive equations. 1. Constitutive equation: definition and basic axioms

Continuum mechanics V. Constitutive equations. 1. Constitutive equation: definition and basic axioms Continuum mechanics office Math 0.107 ales.janka@unifr.ch http://perso.unifr.ch/ales.janka/mechanics Mars 16, 2011, Université de Fribourg 1. Constitutive equation: definition and basic axioms Constitutive

More information

Porous Fibre-Reinforced Materials Under Large Deformations

Porous Fibre-Reinforced Materials Under Large Deformations Porous Fibre-Reinforced Materials Under Large Deformations Salvatore Federico Department of Mechanical and Manufacturing Engineering, The University of Calgary 2500 University Drive NW, Calgary, Alberta,

More information

COMSOL Used for Simulating Biological Remodelling

COMSOL Used for Simulating Biological Remodelling COMSOL Used for Simulating Biological Remodelling S. Di Stefano 1*, M. M. Knodel 1, K. Hashlamoun 2, S. Federico, A. Grillo 1 1. Department of Mathematical Sciences G. L. Lagrange, Politecnico di Torino,

More information

Wave Propagation Through Soft Tissue Matter

Wave Propagation Through Soft Tissue Matter Wave Propagation Through Soft Tissue Matter Marcelo Valdez and Bala Balachandran Center for Energetic Concepts Development Department of Mechanical Engineering University of Maryland College Park, MD 20742-3035

More information

Continuum Mechanics Fundamentals

Continuum Mechanics Fundamentals Continuum Mechanics Fundamentals James R. Rice, notes for ES 220, 12 November 2009; corrections 9 December 2009 Conserved Quantities Let a conseved quantity have amount F per unit volume. Examples are

More information

Lagrangian Formulation of Elastic Wave Equation on Riemannian Manifolds

Lagrangian Formulation of Elastic Wave Equation on Riemannian Manifolds RWE-C3-EAFIT Lagrangian Formulation of Elastic Wave Equation on Riemannian Manifolds Hector Roman Quiceno E. Advisors Ph.D Jairo Alberto Villegas G. Ph.D Diego Alberto Gutierrez I. Centro de Ciencias de

More information

14 - hyperelastic materials. me338 - syllabus hyperelastic materials hyperelastic materials. inflation of a spherical rubber balloon

14 - hyperelastic materials. me338 - syllabus hyperelastic materials hyperelastic materials. inflation of a spherical rubber balloon me338 - syllabus holzapfel nonlinear solid mechanics [2000], chapter 6, pages 205-305 1 2 isotropic hyperelastic materials inflation of a spherical rubber balloon cauchy stress invariants in terms of principal

More information

Waveform inversion and time-reversal imaging in attenuative TI media

Waveform inversion and time-reversal imaging in attenuative TI media Waveform inversion and time-reversal imaging in attenuative TI media Tong Bai 1, Tieyuan Zhu 2, Ilya Tsvankin 1, Xinming Wu 3 1. Colorado School of Mines 2. Penn State University 3. University of Texas

More information

OPENING ANGLE OF HUMAN SAPHENOUS VEIN

OPENING ANGLE OF HUMAN SAPHENOUS VEIN Opening angle of human saphenous vein XIII International Conference on Computational Plasticity. Fundamentals and Applications COMPLAS XIII E. Oñate, D.R.J. Owen, D. Peric and M. Chiumenti (Eds) OPENING

More information

MODELING OF CONCRETE MATERIALS AND STRUCTURES. Kaspar Willam

MODELING OF CONCRETE MATERIALS AND STRUCTURES. Kaspar Willam MODELING OF CONCRETE MATERIALS AND STRUCTURES Class Meeting #1: Fundamentals Kaspar Willam University of Colorado at Boulder Notation: Direct and indicial tensor formulations Fundamentals: Stress and Strain

More information

LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)

LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) RAYTCHO LAZAROV 1 Notations and Basic Functional Spaces Scalar function in R d, d 1 will be denoted by u,

More information

Basic Equations of Elasticity

Basic Equations of Elasticity A Basic Equations of Elasticity A.1 STRESS The state of stress at any point in a loaded bo is defined completely in terms of the nine components of stress: σ xx,σ yy,σ zz,σ xy,σ yx,σ yz,σ zy,σ zx,andσ

More information

Fundamentals of Fluid Dynamics: Elementary Viscous Flow

Fundamentals of Fluid Dynamics: Elementary Viscous Flow Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research

More information

VISCOELASTIC PROPERTIES OF FILLED RUBBER. EXPERIMENTAL OBSERVATIONS AND MATERIAL MODELLING

VISCOELASTIC PROPERTIES OF FILLED RUBBER. EXPERIMENTAL OBSERVATIONS AND MATERIAL MODELLING Engineering MECHANICS, Vol. 14, 2007, No. 1/2, p. 81 89 81 VISCOELASTIC PROPERTIES OF FILLED RUBBER. EXPERIMENTAL OBSERVATIONS AND MATERIAL MODELLING Bohdana Marvalova* The paper presents an application

More information

10 Shallow Water Models

10 Shallow Water Models 10 Shallow Water Models So far, we have studied the effects due to rotation and stratification in isolation. We then looked at the effects of rotation in a barotropic model, but what about if we add stratification

More information

A monolithic fluid structure interaction solver Verification and Validation Application: venous valve motion

A monolithic fluid structure interaction solver Verification and Validation Application: venous valve motion 1 / 41 A monolithic fluid structure interaction solver Verification and Validation Application: venous valve motion Chen-Yu CHIANG O. Pironneau, T.W.H. Sheu, M. Thiriet Laboratoire Jacques-Louis Lions

More information

Course Syllabus: Continuum Mechanics - ME 212A

Course Syllabus: Continuum Mechanics - ME 212A Course Syllabus: Continuum Mechanics - ME 212A Division Course Number Course Title Academic Semester Physical Science and Engineering Division ME 212A Continuum Mechanics Fall Academic Year 2017/2018 Semester

More information

Interaction of Incompressible Fluid and Moving Bodies

Interaction of Incompressible Fluid and Moving Bodies WDS'06 Proceedings of Contributed Papers, Part I, 53 58, 2006. ISBN 80-86732-84-3 MATFYZPRESS Interaction of Incompressible Fluid and Moving Bodies M. Růžička Charles University, Faculty of Mathematics

More information

ELASTOPLASTICITY THEORY by V. A. Lubarda

ELASTOPLASTICITY THEORY by V. A. Lubarda ELASTOPLASTICITY THEORY by V. A. Lubarda Contents Preface xiii Part 1. ELEMENTS OF CONTINUUM MECHANICS 1 Chapter 1. TENSOR PRELIMINARIES 3 1.1. Vectors 3 1.2. Second-Order Tensors 4 1.3. Eigenvalues and

More information